The paper gives a simple proof of Graev’s theorem (asserting that the free product of Hausdorff topological groups is Hausdorff) for a particular case which includes the countable case of $k_\omega $-groups and the countable case of Lindelöf P-groups. For this it is shown that in these particular cases the topology of the free product of Hausdorff topological groups coincides with the $X_0$-topology in the Mal’cev sense, where X is the disjoint union of the topological groups identifying their units.

Given an action ${\varphi }$ of inverse semigroup S on a ring A (with domain of ${\varphi }(s)$ denoted by $D_{s^*}$ ), we show that if the ideals $D_e$ , with e an idempotent, are unital, then the skew inverse semigroup ring $A\rtimes S$ can be realized as the convolution algebra of an ample groupoid with coefficients in a sheaf of (unital) rings. Conversely, we show that the convolution algebra of an ample groupoid with coefficients in a sheaf of rings is isomorphic to a skew inverse semigroup ring of this sort. We recover known results in the literature for Steinberg algebras over a field as special cases.

In this paper, we introduce quotients of étale groupoids. Using the notion of quotients, we describe the abelianizations of groupoid C*-algebras. As another application, we obtain a simple proof that effectiveness of an étale groupoid is implied by a Cuntz–Krieger uniqueness theorem for a universal groupoid C*-algebra.

To study when a paratopological group becomes a topological group, Arhangel’skii et al. [‘Topological games and topologies on groups’, Math. Maced. 8 (2010), 1–19] introduced the class of $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$ -unfavourable spaces. We show that every $\unicode[STIX]{x1D707}$ -complete (or normal) $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$ -unfavourable semitopological group is a topological group. We prove that the product of a $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$ -unfavourable space and a strongly Fréchet $(\unicode[STIX]{x1D6FC},G_{\unicode[STIX]{x1D6F1}})$ -favourable space is $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$ -unfavourable. We also show that continuous closed irreducible mappings preserve the $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$ -unfavourableness in both directions.

We study the locally compact abelian groups in the class ${\mathfrak E_{ \lt \infty }}$ , that is, having only continuous endomorphisms of finite topological entropy, and in its subclass $\mathfrak E_0$ , that is, having all continuous endomorphisms with vanishing topological entropy. We discuss the reduction of the problem to the case of periodic locally compact abelian groups, and then to locally compact abelian p-groups. We show that locally compact abelian p-groups of finite rank belong to ${\mathfrak E_{ \lt \infty }}$ , and that those of them that belong to $\mathfrak E_0$ are precisely the ones with discrete maximal divisible subgroup. Furthermore, the topological entropy of endomorphisms of locally compact abelian p-groups of finite rank coincides with the logarithm of their scale. The backbone of the paper is the Addition Theorem for continuous endomorphisms of locally compact abelian groups. Various versions of the Addition Theorem are established in the paper and used in the proofs of the main results, but its validity in the general case remains an open problem.

We study the following problem: For which Tychonoff spaces $X$ do the free topological group $F(X)$ and the free abelian topological group $A(X)$ admit a quotient homomorphism onto a separable and nontrivial (i.e., not finitely generated) group? The existence of the required quotient homomorphisms is established for several important classes of spaces $X$ , which include the class of pseudocompact spaces, the class of locally compact spaces, the class of $\unicode[STIX]{x1D70E}$ -compact spaces, the class of connected locally connected spaces, and some others.

We also show that there exists an infinite separable precompact topological abelian group $G$ such that every quotient of $G$ is either the one-point group or contains a dense non-separable subgroup and, hence, does not have a countable network.

We prove that $A_{R}(G)\otimes _{R}A_{R}(H)\cong A_{R}(G\times H)$ if $G$ and $H$ are Hausdorff ample groupoids. As part of the proof, we give a new universal property of Steinberg algebras. We then consider the isomorphism problem for tensor products of Leavitt algebras, and show that no diagonal-preserving isomorphism exists between $L_{2,R}\otimes L_{3,R}$ and $L_{2,R}\otimes L_{2,R}$. In fact, there are no unexpected diagonal-preserving isomorphisms between tensor products of finitely many Leavitt algebras. We give an easy proof that every $\ast$-isomorphism of Steinberg algebras over the integers preserves the diagonal, and it follows that $L_{2,\mathbb{Z}}\otimes L_{3,\mathbb{Z}}\not \cong L_{2,\mathbb{Z}}\otimes L_{2,\mathbb{Z}}$ (as $\ast$-rings).

Kaplansky introduced the notions of CCR and GCR $C^{\ast }$ -algebras, because they have a tractable representation theory. Many years later, he introduced the notions of CCR and GCR rings. In this paper we characterize when the algebra of an ample groupoid over a field is CCR and GCR. The results turn out to be exact analogues of the corresponding characterization of locally compact groupoids with CCR and GCR $C^{\ast }$ -algebras. As a consequence, we classify the CCR and GCR Leavitt path algebras.

The Banach–Mazur separable quotient problem asks whether every infinite-dimensional Banach space $B$ has a quotient space that is an infinite-dimensional separable Banach space. The question has remained open for over 80 years, although an affirmative answer is known in special cases such as when $B$ is reflexive or even a dual of a Banach space. Very recently, it has been shown to be true for dual-like spaces. An analogous problem for topological groups is: Does every infinite-dimensional (in the topological sense) connected (Hausdorff) topological group $G$ have a quotient topological group that is infinite dimensional and metrisable? While this is known to be true if $G$ is the underlying topological group of an infinite-dimensional Banach space, it is shown here to be false even if $G$ is the underlying topological group of an infinite-dimensional locally convex space. Indeed, it is shown that the free topological vector space on any countably infinite $k_{\unicode[STIX]{x1D714}}$-space is an infinite-dimensional toplogical vector space which does not have any quotient topological group that is infinite dimensional and metrisable. By contrast, the Graev free abelian topological group and the Graev free topological group on any infinite connected Tychonoff space, both of which are connected topological groups, are shown here to have the tubby torus $\mathbb{T}^{\unicode[STIX]{x1D714}}$, which is an infinite-dimensional metrisable group, as a quotient group.

We show that Matui’s HK conjecture holds for groupoids of unstable equivalence relations and their corresponding $C^{\ast }$-algebras on one-dimensional solenoids.

We compute the homology groups of transformation groupoids associated with odometers and show that certain $(\mathbb{Z}\rtimes \mathbb{Z}_{2})$ -odometers give rise to counterexamples to the HK conjecture, which relates the homology of an essentially principal, minimal, ample groupoid $G$ with the K-theory of $C_{r}^{\ast }(G)$ . We also show that transformation groupoids of odometers satisfy the AH conjecture.

We prove completeness for the main examples of infinite-dimensional Lie groups and some related topological groups. Consider a sequence $G_{1}\subseteq G_{2}\subseteq \cdots \,$ of topological groups $G_{n}$ n such that $G_{n}$ is a subgroup of $G_{n+1}$ and the latter induces the given topology on $G_{n}$, for each $n\in \mathbb{N}$. Let $G$ be the direct limit of the sequence in the category of topological groups. We show that $G$ induces the given topology on each $G_{n}$ whenever $\cup _{n\in \mathbb{N}}V_{1}V_{2}\cdots V_{n}$ is an identity neighbourhood in $G$ for all identity neighbourhoods $V_{n}\subseteq G_{n}$. If, moreover, each $G_{n}$ is complete, then $G$ is complete. We also show that the weak direct product $\oplus _{j\in J}G_{j}$ is complete for each family $(G_{j})_{j\in J}$ of complete Lie groups $G_{j}$. As a consequence, every strict direct limit $G=\cup _{n\in \mathbb{N}}G_{n}$ of finite-dimensional Lie groups is complete, as well as the diffeomorphism group $\text{Diff}_{c}(M)$ of a paracompact finite-dimensional smooth manifold $M$ and the test function group $C_{c}^{k}(M,H)$, for each $k\in \mathbb{N}_{0}\cup \{\infty \}$ and complete Lie group $H$ modelled on a complete locally convex space.

We investigate continuous transitive actions of semitopological groups on spaces, as well as separately continuous transitive actions of topological groups.

We consider the following question: for which metrizable separable spaces $X$ does the free abelian topological group $A(X\oplus X)$ isomorphically embed into $A(X)$ . While for many natural spaces $X$ such an embedding exists, our main result shows that if $X$ is a Cook continuum or $X$ is a rigid Bernstein set, then $A(X\oplus X)$ does not embed into $A(X)$ as a topological subgroup. The analogous statement is true for the free boolean group $B(X)$ .

Answering a longstanding problem originating in Christensen’s seminal work on Haar null sets [Math. Scand. 28 (1971), 124–128; Israel J. Math. 13 (1972), 255–260; Topology and Borel Structure. Descriptive Topology and Set Theory with Applications to Functional Analysis and Measure Theory, North-Holland Mathematics Studies, 10 (Notas de Matematica, No. 51). (North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., Inc., New York, 1974), iii+133 pp], we show that a universally measurable homomorphism between Polish groups is automatically continuous. Using our general analysis of continuity of group homomorphisms, this result is used to calibrate the strength of the existence of a discontinuous homomorphism between Polish groups. In particular, it is shown that, modulo $\text{ZF}+\text{DC}$, the existence of a discontinuous homomorphism between Polish groups implies that the Hamming graph on $\{0,1\}^{\mathbb{N}}$ has finite chromatic number.

We consider generalised metrisability and cardinal invariants in quasitopological groups. We construct examples to show that some equalities of cardinal invariants in topological groups cannot be extended to quasitopological groups.

We extend Følner’s amenability criterion to the realm of general topological groups. Building on this, we show that a topological group $G$ is amenable if and only if its left-translation action can be approximated in a uniform manner by amenable actions on the set  $G$ . As applications we obtain a topological version of Whyte’s geometric solution to the von Neumann problem and give an affirmative answer to a question posed by Rosendal.

The author has previously associated to each commutative ring with unit $R$ and étale groupoid $\mathscr{G}$ with locally compact, Hausdorff and totally disconnected unit space an $R$ -algebra $R\,\mathscr{G}$ . In this paper we characterize when $R\,\mathscr{G}$ is Noetherian and when it is Artinian. As corollaries, we extend the characterization of Abrams, Aranda Pino and Siles Molina of finite-dimensional and of Noetherian Leavitt path algebras over a field to arbitrary commutative coefficient rings and we recover the characterization of Okniński of Noetherian inverse semigroup algebras and of Zelmanov of Artinian inverse semigroup algebras.

Based on the abstract version of the Smital property, we introduce an operator $DS$ . We use it to characterise the class of semitopological abelian groups, for which addition is a quasicontinuous operation.

A class of abelian topological groups was previously defined to be a variety of topological groups with coproducts if it is closed under forming subgroups, quotients, products and coproducts in the category of all abelian topological groups and continuous homomorphisms. This extended research on varieties of topological groups initiated by the second author. The key to describing varieties of topological groups generated by various classes was proving that all topological groups in the variety are a quotient of a subgroup of a product of groups in the generating class. This paper analyses generating varieties of topological groups with coproducts. It focuses on the interplay between forming products and coproducts. It is proved that the variety of topological groups with coproducts generated by all discrete groups contains topological groups which cannot be expressed as a quotient of a subgroup of a product of a coproduct of discrete groups. It is proved that the variety of topological groups with coproducts generated by any infinite-dimensional Hilbert space contains all infinite-dimensional Hilbert spaces, answering an open question. This contrasts with the result that a variety of topological groups generated by a topological group does not contain any infinite-dimensional Hilbert space of greater cardinality.